AP Calculus AB Exam Question

Let f(x) and g(x) be differentiable functions on an interval containing x = a, except possibly at x = a. Suppose that f(a) = g(a) = 0 and that g'(a) ≠ 0. Use L'Hôpital's Rule, if applicable, to determine the value of the following limit:

Answer with Step-by-Step Explanation

To solve this problem, we will use L'Hôpital's Rule, which states that if the limit of the ratio of two functions exists in an indeterminate form (such as 0/0 or ∞/∞), then the limit is equal to the limit of the ratio of their derivatives.

Given that f(a) = g(a) = 0 and g'(a) ≠ 0, we can rewrite the limit as $\lim_{x\to a} \frac{f(x) - f(a)}{g(x) - g(a)}$. Applying L'Hôpital's Rule, we take the derivative of both the numerator and denominator:

Since f(x) and g(x) are both differentiable, we can evaluate the limit as x approaches a:

Therefore, the value of the limit is equal to $\frac{f'(a)}{g'(a)}$.

Note: L'Hôpital's Rule can only be applied when the limit is an indeterminate form (0/0 or ∞/∞). It is always a good practice to check the conditions for applying L'Hôpital's Rule before using it. Additionally, if the limit is not in an indeterminate form, L'Hôpital's Rule does not apply.